Density functional theory with Wigner distributions · Density functional theory with Wigner...

Phase-space quasidensities: the basicsAtomic Wigner functions

Natural Wigner orbitalsCoda: fixing a broken egg

Density functional theory with Wignerdistributions

J. M. Gracia–Bondıa

Universidad de Zaragoza

joint work w/ Philippe Blanchard (Bielefeld) andJoseph C. Varilly (San Jose)

ZiF, Bielefeld, 28 Apr 2012

J. M. Gracia–Bondıa Density functional theory with Wigner distributions



1 Phase-space quasidensities: the basics

2 Atomic Wigner functions

3 Natural Wigner orbitals

4 Coda: fixing a broken egg




IntroductionThe Hohenberg-Kohn-Levy theorem

QM of electronic systems

Thanks to Volker’s lecture, I may go straight to the goal. ;D

We deal with the nonrelativistic QM of an electronic system:

H = T + Vext + Vee =N∑j

−1

2∆~qj

+ V (~qj) +N∑i<j

1

|~qi −~qj |.

Pure states DN = |Ψ〉〈Ψ| have skewsymmetric Ψ(x1, . . . , xN) withxj = (~qj , ςj): spatial and spin variables.

Integrating out x3, . . . , xN gives a reduced 2-matrix:

D2(x1, x2; x ′1, x′2) =

∫DN(x1, . . . , xN ; x ′1, x

′2, x3, . . . , xN) dx3 . . . dxN .

We can integrate further to get the reduced 1-matrix:

D1(x ; x ′) =

∫D2(x , x2; x ′, x2) dx2.





Reduced 1-matrices

Let now think of ions, and define γ = (N − 1)/Z . There is thehelium-like energy functional:

E(D2) =NZ 2

2Tr

[(−

∆~q1+ ∆~q2

2− 1

|~q1|− 1

|~q2|+

γ

|~q1 −~q2|

)D2

].

The ground-state energy minimizes E(D2); but over what set of2-matrices, we do not know.This is the N-representability problem.

(Also D1 plus the pair density would do.)

The set of admissible 1-matrices is indeed known [Coleman 1963]:D1 ≥ 0, trD1 = 1, and all eigenvalues of D1 obey 0 ≤ λ ≤ 1/N. Itis a convex set and its extremal states are those coming fromHartree–Fock states.





Illustrious ancestors

The electron density comes from the diagonal, summed over spinstates:

ρ(~q) := N∑

ς D1(~q, ς;~q, ς).

Note the normalization∫ρ(~q) d~q = N. This is the variable in

Kohn’s DFT, which tries to avoid the “exponential wall” faced byFCI calculations.

At any rate, much info about the system depends on ρ only. Forinstance, the Thomas–Fermi theory (1927-28) approximates thekinetic energy as the functional TTF[ρ] = CF

∫ρ5/3(~q) d~q, with

CF.

= 2.871.

D1-functional theory is natural and looks as a good compromise!





Enter Wigner functions

Henceforth when possible I replace DN by a phase-space function

PN(q; p) ≡ PN(~q1, . . . ,~qN ;~p1, . . . ,~pN),

omitting spin indices for notational simplicity. This is the Wignerfunction for |Ψ〉〈Ψ|:

PN(q; p) := π−3N

∫DN(q − z ; q + z) e2ip·z d~z1 · · · d~zN .

This is real, and TrDN = 1 becomes∫PN(q; p) = 1. However,

PN can take perhaps some negative values.

One can also model transitions |Ψ〉〈Φ| by

PΨΦ(q; p) := π−3N

∫Ψ(q − z)Φ∗(q + z) e2ip·z d~z1 . . . d~zN .





1 and 2-body quasidensities

Now D2 and D1 are replaced by quasidensities:

d2(~q1,~q2,~p1,~p2) =

(N

2

)∫PN(q; p) d~q3 . . . d~qN d~p3 . . . d~pN ,

d(~q,~p) := d1(~q,~p) = N

∫PN(q; p) d~q2 . . . d~qN d~p2 . . . d~pN .

With these normalizations, ρ(~q) =∫d1(~q,~p) d~p.

Operator products FG ←→ Moyal products f ? g . They obey∫f ? g(~q,~p) d~q d~p =

∫f (~q,~p) g(~q,~p) d~q d~p.





Variational principle

Spectral theory works fine in the Moyal product notation. Assumethat H has purely discrete, nondegenerate spectrum,E0 < E1 < · · · < En < · · · . One solves the eigenvalue problems:

H ? Γnm = Em Γmn, Γmn ? H = En Γmn,

to get an orthonormal basis of states Γnn and transitions Γmn,m 6= n, on phase space. They obey Γmn ? Γrs = δnrΓms .

Then H =∑

n En Γnn, so the variational principle kicks in:

E0 ≤∫

H(q; p)PN(q; p) dq dp,

with equality if and only if PN = Pgs, the Wigner function for theground state.





Ground-state quasidensity

HKL-type theorem: The ground state quasidensity dgs(~q;~p)determines the many-body ground state.

Proof. Recall H = T + Vee + Vext. If two different externalpotentials Vext, V

′ext had the same dgs, that would mean

E ′0 =

∫H ′P ′gs dq dp <

∫H ′Pgs dq dp

= E0 +

∫(V ′ext − Vext)Pgs dq dp,

but Vext is a 1-particle function, so that∫(V ′ext − Vext)Pgs dq dp =

∫(V ′ext − Vext)(~q;~p) dgs(~q;~p) d~q d~p.

Exchanging Vext and V ′ext then gives an impossibility:

E0 − E ′0 <

∫(Vext − V ′ext) dgs d~q d~p < E0 − E ′0.





The energy functional

Theorem: There is a functional A[d ], indepnt of Vext, such that

E0 ≤ A[d ] +

∫( 1

2 |~p|2 + V (~q;~p)) d(~q;~p) d~q d~p =: E[d ],

with equality when d = dgs.

Indeed, one can define A[d ] by

A[d ] := min{∫

P(q; p)Vee(q; p) dq dp : P 7→ d}

ranging over all Wigner N-functions P with 1-quasidensity d(~q;~p).

The functional A[d ] attains its minimum at d = dgs. This wouldbe all we need, if we knew a concrete form for A[d ]. But, we don’t.

An advantage over DFT: we know the kinetic energy functional, soour A[d ] is better than their E[ρ].





Scaling

A[d ] is better than E[ρ] for many reasons.

Scale the Wigner distribution by defining Pλ(q; p) = Pλ(λq;λ−1p),implying dλ(~q;~p) = d(λ~q;λ−1~p). This dλ corresponds to aHamiltonian of the form T + λVee +

∑Ni=1 λ

2Vd(~qi ;~pi ). Now:

A[dλ] = λA[d ]; and trivially∂A[dλ]

∂λ

∣∣∣∣λ=1

= A[d ].

The situation in standard DFT approach with respect to scaling ismuch more involved: the naive expectations T [ρλ] = λ2T [ρ] andVee[ρλ] = λVee[ρ] for the summands of the universalHohenberg–Kohn–Levy functional are both false.

Last for now, but not least: the Thomas–Fermi model finds hereits long home at last.




Gaussian basis setsGaussians states and transitions in phase spaceWhat atomic Wigner functions look like

Using quasidensities

Consider a 1-electron atom with Hamiltonian, in atomic units:

H(~q,~p) =|~p|2

2− Z

|~q|We know the ground-state wave-function:

ψ1s(~q) =Z 3/2

√π

e−Zr ; ψ1s(~p) =2√

2

π

Z 5/2

(Z 2 + |~p2|)2.

Unfortunately, a closed form for the Wigner function:

P1s(~q;~p) =Z 3

π4

∫e−Z |~q−~z|e−Z |~q+~z|e−2i~p·~z d~z .

is not available.However, with two or more electrons, even wave functions becomehard to handle.





Gaussian basis sets

So routinely chemists expand wave functions in Gaussian basis. Forhydrogen:

ψ(~q) = (α/π)3/4e−12αr2, P(~q,~p) = π−3 e−αr

2−p2/α.

We know that E0 <∫HP dq dp, so

E0 <1

π3

∫ (p2

2− Z

r

)e−αr

2−p2/α d~q d~p =3α

4− 2Z

√α√π

,

which is minimized at α0 = 16Z 2/9π, yielding the approximation

E = −4Z 2

3π= −Z 2

2

(8

3π

)> −Z 2

2= E0.

With two Gaussians, P = c21P1 + c2

2P2 + c1c2(P12 + P21), the bestfit gives E

.= −0.4858Z 2. With three Gaussians, one reaches

E.

= −0.496979Z 2. This is the heart of the STO-3G method: “use3 Gaussians for each Slater-type orbital”.





Gaussian Wigner functions

In general, normalized Gaussians centred at (0, 0) look like

PF (q, p) = π−3N√

detF e−(q,p)F (q,p)t .

Here F is a positive definite matrix: F > 0.

When P ↔ |Ψ〉〈Ψ| is a pure state, F is also symplectic [Littlejohn1986]: FJF t = J, with the standard complex structure. Thus wecan write

F =

(A BBt D−1

); D > 0, BD = DBt , A = D + BDBt .

The corresponding wave function is Ψ(q) ∝ e−12

(q·Dq+iq·BDq).

Question: If Ψ1 and Ψ2 are Gaussian, and P12 ↔ |Ψ1〉〈Ψ2|, whatdoes the transition function P12 look like?





Gaussian transitions

Put D12 := 12 (D1 + D2 + i(B1D1 − B2D2)), so that

〈Ψ2 |Ψ1〉 ∝∫

e−q·D12q dq with <(D12) > 0.

Now define B12 by

B12D12 := − i2 (D1 − D2) + 1

2 (B1D1 + B2D2);

and lastly, set A12 := D12 + B12D12Bt12.

Lemma: the Gaussian transition P12 is given by

P12(q, p) ∝ e−(q,p)F12(q,p)t where F12 =

(A12 B12

Bt12 D−1

12

),

i.e., F12 is symmetric and complex symplectic, with <F12 > 0.

¿Do all such Gaussians represent transitions? A good question.





The radial functions

For a closer look at the atomic Wigner function P(~q,~p), weintegrate out the angle θ between ~q and ~p, to get a first-quadrantfunction of the radii:

G (r , p) := 8π2r2p2

∫ π

0P(~q,~p) sin θ dθ.

For a single Gaussian, this is just (4π2r2e−αr2)(4π2p2e−p

2/α),which is everywhere positive with a maximum at (α−1/2, α1/2).

The true G (r , p) for the H-atom dips into negative territory, withsmall oscillations around 0 for large r and p.

We next show a few computed G (r , p) for hydrogen and for someclosed-shell atoms.





Atomic WFs: Hydrogen

We see small amplitudeoscillations in theasymptotic region oflarge r and p.





Atomic WFs: Helium

Note the mildoscillationsbeyond theThomas–Fermicurve.





Atomic WFs: Beryllium

Note the central hole dueto interference betweenthe 1s and 2s electrons.





Atomic WFs: Neon

Note the dipnear the origindue to the 2pshell.




SLK functional(s) for 2-electron systemsNatural Wigner orbitals

The orbital expansion

If dN(q, p) is a pure state, its quasidensity d1 can be written as

d1(~q,~p; ς, ς ′) =∑

r≥1nr frr (~q,~p; ς, ς ′),

where the frr are orthonormal,∑

r nr = N and 0 ≤ nr ≤ 1 byColeman’s theorem: the ni are occupation numbers.

For closed shells, N is even, and d1(~q,~p) =∑

r≥1 2nr frr (~q,~p).

The Wigner natural orbitals frr (~q,~p)↔ |φr 〉〈φr | are part of anorthonormal basis, including transitions frs(~q,~p)↔ |φr 〉〈φs |, suchthat fjk ? frs = δkr fjs and

d1 ? frs = 2nr frs ; frs ? d1 = 2ns frs .





Two-electron systems

For 2-electron systems, like H−, He, Li+, H2, one can say more.There are singlet states, Ψ(x1, x2) = 2−1/2(↑1↓2 − ↓1↑2)ψ(~q1,~q2)with ψ symmetric, or triplet states with spin-symmetric,spatial-skew wave functions. Here the Wigner function is P2 ≡ d2.

Expanding the singlet state for d2, and tracing over spins, it comes

d2(~q1,~q2;~p1,~p2) =∑

r ,s≥02crcs frs(~q1,~p1)frs(~q2,~p2),

in a suitable basis of natural Wigner orbitals/transitions frs .

Since∫frs d~q d~p = δrs , we get d1 =

∑r≥0 2c2

r frr . Thus c2r = nr .

This means that d2 could be recovered from d1 once we know thebasis {frs}, if only we knew the signs of cr = ±√nr . This is thephase dilemma [Lowdin-Shull 1956, Kutzelnigg 1963].





SLK energy functional

Let us abbreviate frs(~q) := frs(~q,~p) d~p. The contribution of frs tothe interaction energy is

Lrs := 2

∫frs(~q1)frs(~q2)

|~q1 −~q2|d~q1 d~q2.

The Shull-Lowdin-Kutzelnigg (SLK) functional for 2-electronsystems is then

AL[d ] :=∑

r ,s≥0crcs Lrs =

∑r ,s≥0

±√nrns Lrs .

For the correct choice of signs, this is exact: AL[d ] = A[d ]!But, what is that correct choice? We take c0 = +

√n0, for free.

For ions, a good empirical choice is then cr = −√nr for r ≥ 1.However, for the H2 molecule, some more + signs are required.





Enter Muller

The interaction energy functional A[d ] can be approximated byone of the form “Coulomb” + “exchange”:

AHF[d ] =1

2

∫ρ(~q1)ρ(~q2)

|~q1 −~q2|d~q1 d~q2 −

1

2

∫|γ(x1, x2)|2

|~q1 −~q2|dx1 dx2

where the γ(x1, x2) is a known functional; this is exact only forSlater determinants. The term |γ(x1, x2)|2 can be written, in theclosed-shell 2-electron case, as∑

r ,snrns frs(~q1) fsr (~q2).

Muller (1984) proposed to replace each nrns by√nrns , thereby

lowering the energy estimate. For N = 2, a complicated analysis[Frank, Lieb, S & S, 2007] shows that the new estimate is too low:AM[d ] < A[d ].

It turns out that this follows from a simple comparison argument.J. M. Gracia–Bondıa Density functional theory with Wigner distributions




SLK versus Muller

We compare the pair density of the Muller functional with that ofa typical SLK functional. These are:

ρ2M(~q1,~q2) :=1

2ρ(~q1)ρ(~q2)

−∑r≥0

nrρr (~q1)ρr (~q2)−∑r 6=s

√nrns frs(~q1)frs(~q2); and

ρ2L(~q1,~q2) := 2∑r≥0

nr ρr (~q1)ρr (~q2)

− 2∑r≥1

√n0nr f0r (~q1)f0r (~q2) +

∑r 6=s,≥1

√nrns frs(~q1)frs(~q2).

Now we ask: how do they compare,

A =1

2ρ(~q1)ρ(~q2) or B = 2

∑r≥0

nr ρr (~q1)ρr (~q2) ?





Bean counting

Since the occupation numbers satisfy∑

r nr = 1, and thereforen2r = nr −

∑s 6=r nrns for each r , the difference is

B − A =∑

r 6=snrns [frr (~q1)− fss(~q1)] [frr (~q2)− fss(~q2)].

Since the Coulomb potential is of positive type, integration gives∫(RHS)

|~q1 −~q2|d~q1 d~q2 > 0,

which in turn implies that AM[d ] < AL[d ].

To write ρ2L, we used a particular choice of signs; but any otherchoice alters only the summations that we were able to ignore.

Thus we conclude that AM[d ] < A[d ] for N = 2: the Mullerfunctional is indeed overbinding.




The simple-minded harmonium model

The phase-space formalism is ideally suited to handle the “fakeatom” harmonium [Moshinsky 1968]:

H =1

2(p2

1 + p22) +

ω2

2(r2

1 + r22 )− k

4r212, where 0 < k < ω2.

With µ =√ω2 − k, using the extracule ~R := (~q1 +~q2)/

√2 and

intracule ~r := (~q1 −~q2)/√

2, we get two independent oscillators:

H = HR + Hr = 12 (P2 + ω2R2) + 1

2 (p2 + µ2r2),

whose ground state is d2 = π−6e−2HR/ωe−2Hr/µ, a pure-stateGaussian.

Integrating over (~q2,~p2) gives the 1-quasidensity:

d1(~q,~p) =

(2√ωµ

π(ω + µ)

)3/2

e−2(p2+ωµr2)/(ω+µ).




A familiar 1-density

This d1 is Gaussian, but of the mixed-state kind. Indeed, reducingto one dimension, write λ := 2

√ωµ/(ω + µ), and put

Q := (ωµ)1/4q, P := (ωµ)−1/4p, we obtain

d1(q, p) =λ

πe−λ(Q2+P2).

This has the form of a Gibbs state: if λ =: tanhβ/2, we expand itas

d1(q, p) =2

πsinh

β

2

∑r≥0

frr (Q,P) e−(r+ 12

)β,

where these frs are the (relatively) well-known phase-spaceeigentransitions for a harmonic oscillator; the diagonal ones frr aregiven by Laguerre functions.




Humpty Dumpty returns

One can solve the sign dilemma explicitly here. We only need toresum the SLK functional

∞∑r ,s=0

±rs√nrns frs(~q1,~p1) frs(~q2,~p2)

with nr = n0 e−rβ; and to compare it with the known d2.

Summing over sub-diagonals, after some juggling with Besselfunctions, one finds an exact match if and only if the signs follow achessboard pattern:

±rs = (−1)r+s .

Thus the phase ambiguity has been resolved analytically and for thefirst time the mixed state d1 has been unscrambled to reassemblethe pure state d2. (This is harder to do when not working on phasespace. Minimization of course works, but it is less fun.)




Harmonium as a laboratory bench

In the last few years, the model has been used extensively to study(Lowdin’s) electronic correlation, approximation of functionals andquestions of entanglement (provided by the nr ). Entropy correlateswell with electronic correlation. The pioneer paper was bySrednicki, to argue about black hole entropy (1993).

Something fun: the Muller functional gives exactly the same result(Benavides, Nagy and collaborators) for the energy on the “true”states as the exact one (so it also overbinding here).

The Goedecker–Umrigar, Csanyi–Goedecker–Arias,Buijse–Baerends and other functionals perform worse for thismodel than the Muller functional.




Briefly, on the first excited state

For one mode:

d2(R, r ;P, p) =2

π2exp

(−2HR

ω

)exp

(−2Hr

2µ

)(p2 + µ2x2

µ− 1

2

).

The reduced one-body spinless quasidensity is:

d1(r ; p) = 2

∫d2(r , r2; p, p2) dr2 dp2

=2

π

(2√ωµ

ω + µ

)3

e−2 ωµω+µ

r2−2 1ω+µ

p2(ωr2 +

1

ωp2

),

an interesting positive, non-Gaussian Wigner function representinga mixed state.




A look at the occupation numbers

(Numerical)

Only the 2ndoccupationnumber evergrows large. . .




A look at the occupation numbers II

. . . and less sothan for thesinglet state.

There is no “sign dilemma” from the occupation numbers of thespin triplet system; but only an ambiguity in the choice of naturalorbitals.


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